Deriving the Golden Ratio

"A straight line is said to have been cut in extreme and
mean ratio when, as the whole line is to the greater segment,
so is the greater to the lesser."

Euclid, ca. 300 BC

    

Two distances are said to be at the golden ratio if the ratio of their sum to the greater distance is equal to the ratio of the greater to the lesser. There are countless rich examples of how this ratio describes the proportions of things in nature and how it was used by many artists and architects to create aesthetically pleasing works.

       

We can derive the actual value by considering a line segment whose parts obey the golden ratio. By starting with the “mean and extreme ratios,” we can solve for a, using the quadratic formula, and then solve for the ratio of a to b.

We can reject the negative value since we’re working with measurements and the ratio between positive numbers has to be positive. But since 0.618 is the reciprocal of 1.618, it still plays an interesting role.

                   

A final interesting piece is that if you start with a golden rectangle, you can divide it into a square and a smaller golden rectangle, and then again into a smaller square and an even smaller golden rectangle, etc. And if you keep going, what you find is that the proportions of each new golden rectangle follow the Fibonacci sequence.

         

It turns out that the ratio of two consecutive numbers in the Fibonacci sequence approaches the golden ratio as the number in the sequence approaches infinity.

The math here is very interesting, and there’s plenty more to explore!